Properties

Label 2-85176-1.1-c1-0-36
Degree $2$
Conductor $85176$
Sign $-1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 11-s − 3·17-s + 19-s − 3·23-s − 4·25-s − 3·29-s − 2·31-s − 35-s + 5·37-s − 10·41-s + 5·43-s − 8·47-s + 49-s + 6·53-s + 55-s − 6·59-s + 9·61-s − 8·67-s − 2·71-s − 13·73-s − 77-s + 10·79-s + 14·83-s + 3·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s − 0.727·17-s + 0.229·19-s − 0.625·23-s − 4/5·25-s − 0.557·29-s − 0.359·31-s − 0.169·35-s + 0.821·37-s − 1.56·41-s + 0.762·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s − 0.781·59-s + 1.15·61-s − 0.977·67-s − 0.237·71-s − 1.52·73-s − 0.113·77-s + 1.12·79-s + 1.53·83-s + 0.325·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.29017327251416, −13.54066281187426, −13.23833050914090, −12.80620222406501, −11.95096354868633, −11.78999941910130, −11.27111579967220, −10.77018090434124, −10.12027422054136, −9.837734390238070, −8.901662172905950, −8.821723793639422, −7.976376832434206, −7.622758231282896, −7.238962056551019, −6.360701687908123, −6.066478001353368, −5.291856706684053, −4.795425832372854, −4.238124360135977, −3.624625153170827, −3.110843898144685, −2.127790052269477, −1.870895231256841, −0.7797209551216649, 0, 0.7797209551216649, 1.870895231256841, 2.127790052269477, 3.110843898144685, 3.624625153170827, 4.238124360135977, 4.795425832372854, 5.291856706684053, 6.066478001353368, 6.360701687908123, 7.238962056551019, 7.622758231282896, 7.976376832434206, 8.821723793639422, 8.901662172905950, 9.837734390238070, 10.12027422054136, 10.77018090434124, 11.27111579967220, 11.78999941910130, 11.95096354868633, 12.80620222406501, 13.23833050914090, 13.54066281187426, 14.29017327251416

Graph of the $Z$-function along the critical line